Pacific Northwest Numerical Analysis Seminar
October 9, 1999

Time Splitting and Time Stepping Methods for Ocean Circulation Modeling
Robert L. Higdon

Numerical models of ocean circulation typically admit motions varying on a
wide range of time scales. These include external gravity waves, which can
travel with speeds of up to hundreds of meters per second. On the other hand,
motions such as currents and internal gravity waves exhibit speeds on the
order of a meter per second or less. In the case of external waves, the
disturbances are nearly independent of depth, and for the sake of efficiency
it is common practice to represent these motions with a two-dimensional
subsystem which is solved by techniques that are different from those used to
simulate the remaining three-dimensional motions. However, if this splitting
of fast and slow time scales is insufficiently accurate, then the resulting
algorithm can be unstable. The instability can be removed by using a more
precise vertical averaging when deriving the two-dimensional system that
models the fast waves. One goal of this talk is to outline these results.

A second goal is to describe some other issues related to time stepping
schemes. A time discretization that is commonly used in geophysical
fluid dynamics is the leap-frog method, which is a three-level scheme based on
centered differencing. One drawback of this method is a computational mode
consisting of nonphysical, grid-scale oscillations with respect to time.
These oscillations contaminate the solution and can be strongly stimulated by
impulsive forcing. This is the case, for example, with a prominent ocean
model which allows sudden transfers of mass between the uppermost ``mixed''
layer and the stratified interior. These problems can be avoided by using a
time stepping method that uses only two time levels. I will sketch some
preliminary results for a two-level scheme which incorporates the improved
splitting mentioned above.